Question

ap stats: normal distribution calculations - practice
mr. wilcox spends a full day at gingerman raceway testing some new tires on his honda civic. for 400 laps, he records his lap time in minutes. at the end of the day, he plots the distribution of times and realizes that it follows an approximately normal distribution with a mean of 1.84 and a standard deviation of 0.07.

1. label the values 1, 2, and 3 standard deviations above and below the mean.
2. what percent of the lap times are less than 1.70 minutes?
3. what percent of the lap times are greater than 1.77 minutes?
4. what percent of the lap times are between 1.70 and 1.91 minutes?
5. what lap time would be in the slowest 2.5% of all lap times?

Explanation:

Step1: Calculate z-scores for each value

For a value $x$, $z = \frac{x - \mu}{\sigma}$, where $\mu=1.84$, $\sigma=0.07$

• $-3\sigma$: $1.84 - 3(0.07) = 1.63$
• $-2\sigma$: $1.84 - 2(0.07) = 1.70$
• $-1\sigma$: $1.84 - 1(0.07) = 1.77$
• $\mu$: $1.84$
• $+1\sigma$: $1.84 + 1(0.07) = 1.91$
• $+2\sigma$: $1.84 + 2(0.07) = 1.98$
• $+3\sigma$: $1.84 + 3(0.07) = 2.05$

Step2: Solve Q2: % < 1.70

1.70 is $-2\sigma$. From empirical rule, % below $-2\sigma$ is $2.5\%$.

Step3: Solve Q3: % > 1.77

1.77 is $-1\sigma$. Total % above $-1\sigma$: $50\% + 34\% = 84\%$.

Step4: Solve Q4: % between 1.70 & 1.91

1.70 = $-2\sigma$, 1.91 = $+1\sigma$. Sum intervals: $13.5\% + 34\% + 34\% = 81.5\%$.

Step5: Solve Q5: Slowest 2.5% lap time

Slowest 2.5% is $+2\sigma$: $1.84 + 2(0.07) = 1.98$.

Answer:

1. Labeled values (left to right): 1.63 ($-3\sigma$), 1.70 ($-2\sigma$), 1.77 ($-1\sigma$), 1.84 (mean), 1.91 ($+1\sigma$), 1.98 ($+2\sigma$), 2.05 ($+3\sigma$)
2. $2.5\%$
3. $84\%$
4. $81.5\%$
5. 1.98 minutes